On Geodesic E-Convex Sets, Geodesic E-Convex Functions and E-Epigraphs
- 454 Downloads
In this paper, we introduce a new class of sets and a new class of functions called geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold. The concept of E-quasiconvex functions on R n is extended to geodesic E-quasiconvex functions on Riemannian manifold and some of its properties are investigated. Afterwards, we generalize the notion of epigraph called E-epigraph and discuss a characterization of geodesic E-convex functions in terms of its E-epigraph. Some properties of geodesic E-convex sets are also studied.
KeywordsGeodesic E-convex sets Geodesic E-convex functions Geodesic E-quasiconvex function E-epigraphs Riemannian manifolds
The authors are highly thankful to anonymous referees and the editor for their valuable suggestions/comments which have contributed to the final preparation of the paper.
- 1.Arana, M., Ruiz, G. Rufian, A. (eds.): Optimality Conditions in Vector Optimization. Bentham Science, Bussum (2010) Google Scholar
- 3.Danzer, L., Gruenbaum, B., Klee, V.: Helly’s theorem and its relatives. In: Klee, V. (ed.) Convexity. Proc. Sympos. Pure Math., vol. 7, pp. 101–180. Amer. Math. Soc., Providence (1963) Google Scholar
- 7.Zalinescu, C.: A critical view on invexity. J. Optim. Theory Appl. (2011). (To appear) Google Scholar
- 14.Duca, D.I., Duca, E., Lupsa, L., Blaga, R.: E-convex functions. Bull. Appl. Comput. Math. 43, 93–103 (2000) Google Scholar
- 19.Rapcsak, T.: Smooth Nonlinear Optimization in ℝn. Kluwer Academic, Amsterdam (1997) Google Scholar
- 24.Ahmad, I., Iqbal, A., Ali, S.: On properties of geodesic η-preinvex functions. Adv. Oper. Res., 10 pp. (2009). Article ID 381831 Google Scholar